NOTE: This
page is a continuation of the notes and worksheets for topic 9.7 Astrophysics.
Four separate pages were used for this topic because of the large volume of
material in the topic. This will keep download time within acceptable
limits.

BINARY
STARS

NOTE:
Numbers appearing in parentheses at the end of sentences or paragraphs refer to
the references provided in the Bibliography at the end of these notes.

The only direct way to determine
the mass of a star is by studying its gravitational interaction
with other objects.Binary
stars, two stars in orbit around a common centre of mass, provide an ideal
opportunity to do just that.At
least half of all the “stars” in the sky are actually multiple star systems.Analysis of the orbital parameters of such systems provides vital
information about a variety of stellar characteristics, including mass.(1)

The methods used to analyse the
orbital data depend on the geometry of the system, its distance from the
observer and the relative masses and luminosities of each component.As a consequence, binary star systems are classified according to the
means by which they are detected.(1)

There
are six main types of binary systems: visual, spectroscopic, eclipsing,
astrometric, spectrum and optical (1).The first four of these are required for study by the current syllabus.Let us now describe these four different types of binary star systems.

When astronomers can actually
see the two stars orbiting each other, the binary is called a visual binary.The brighter star in the pair is called the primary star and is denoted
by the letter A after its name, while the other star is called the secondary and
is denoted by the letter B.For
example, the alpha star in the constellation of the Southern Cross is actually a
visual binary consisting of Alpha Crucis A and Alpha Crucis B, both of which are
easily discernable through a small telescope.

Many years of careful observation
are often necessary to ensure that the two stars are actually in orbit around
each other.If the stars do form a
binary system, each star will follow an elliptical orbit around the centre of
mass of the system.The star with
the larger mass will stay closer to the centre of mass of the system and will
therefore have the smaller orbit.This
makes sense – just think of two children on a seesaw.To balance, the child with the greater mass must sit closer
to the fulcrum.

Some binary systems are
detected due to a periodic shift in spectral lines.These are called spectroscopic binaries and are detected using the
Doppler Effect.If two stars in
a binary system have some component of their orbital motion along the
observer’s line of sight, then as the stars move around their orbits, they
periodically approach and recede from the observer.Thus, the spectral lines of the two stars are alternately blueshifted and
then redshifted.(1 & 3)

Through a telescope such a star
system may appear to be one single star.However,
an examination of the spectra coming from “the star” will reveal two sets of
spectral lines, one set from each star, each of which will display Doppler
shifts.The two overlapping spectra
shift relative to each other due to the Doppler Effect.When the spectrum of each star in the binary system is visible, the
binaries are called double-line spectroscopic binaries.If, however, one star in the pair is much more luminous than the other,
the spectrum of the less luminous companion may be overwhelmed and only a single
set of spectral lines will be seen.In
this case, the spectral lines shift back and forth due to the Doppler Effect and
therefore still reveal the binary system.This
type of binary is called a single-line spectroscopic binary.(1 & 3)

Consider the diagram below, which
I have adapted from Reference 1 p.204.

The diagram above shows the
periodic shift in spectral features of a double-line spectroscopic binary.For simplicity, only one spectral line from each star is shown.Note that l0
is the non-Doppler shifted wavelength of the lines and wavelength increases to
the right of that position.The
relative wavelengths of the spectra of stars 1 and 2 are shown at four different
phases during orbit: (a) star 1 is moving towards Earth while star 2 is moving
away; (b) both stars have velocities perpendicular to the line of sight; (c)
star 1 is receding from Earth and star 2 is approaching Earth; and (d) again
both stars have velocities perpendicular to the line of sight of the observer on
Earth.

Note that the spectroscopic
detection of binaries is most likely if the period of the motion is short and
the orbital speeds of the stars are high.So,
most spectroscopic binaries are close binary systems.(3)

Some binaries have their orbital
planes oriented approximately along the line of sight of the observer.In this case, one star may periodically pass in front of the other,
blocking light from the eclipsed star.Such
a system is called an eclipsing binary and is recognized by regular variations
in the amount of light received at the telescope (1).Light curves (plots of apparent visual magnitude versus time or
orbital phase) can be recorded for eclipsing binaries using an appropriate
photometer.Such curves can reveal
enormous amounts of information about the binary system.For example, light curves can confirm the presence of two stars.The ratio of surface temperatures can be determined from how much their
combined light is diminished during eclipse.The duration of the eclipse provides information on the relative radii of
the stars and their orbits.(3)

Further to this, if the
eclipsing binary also happens to be a double-line spectroscopic binary, an
astronomer can calculate the actual mass and radius of each star in the system
from the light curves and the velocity curves.(3)

Examine the light curve
shown below.It is typical of a
binary system consisting of small hot star and a large cool star with the
orbital plane edge-on to our line of sight.Note that the smaller star is travelling from left to right across the
front of the larger star.This
diagram was adapted from Reference 12 p.323.

Note:
Try out this java applet, which allows you to model the light curves of
eclipsing binaries using a computer simulation (Syllabus point 9.7.5, column 3,
dot point 1).

Sometimes one member of a binary
is significantly brighter than the other and may be the only one of the pair
visible through a telescope.In
such a case the existence of the unseen member may be deduced by observing the
oscillatory motion of the visible star (1).This reveals itself as a detectable “wobble” in the star’s proper
motion (5).Since Newton’s First
Law requires that a constant velocity be maintained by a mass unless an external
unbalanced force is acting upon it, such oscillatory motion requires that
another mass be present.Such a
binary star system is called an astrometric binary.(1)

DETERMINATION
OF STELLAR MASSES

As mentioned previously binary
stars are very important in determining the masses of stars.Binary stars orbit each other in accordance with Kepler’s Third Law –
the Law of Periods.For any binary
system we can write:

where m1 = mass of star 1, m2
= mass of star 2, r
= average distance of separation of the two stars, T = orbital period of the binary system and G
= the Universal Gravitational Constant = 6.673 x 10-11 SI.

This equation is known as
Newton’s form of Kepler’s Third Law.Clearly,
for any binary system if we can determine the orbital period and the average
distance of separation of the stars in the system, we can determine the total
mass of the system (m1
+ m2).

For visual binaries we can
usually determine the orbital period, although in some cases observations
over more than one lifetime may be needed.If the distance from Earth to the binary can be calculated from
parallax measurements or using spectroscopic parallax, then the angular
separation between the stars in the binary can be translated into the physical
distance between the stars.Astronomers
must also take into account the angle of tilt of the binary’s orbit to our
line of sight.Substituting into
Kepler’s Third Law equation then allows us to calculate the total mass of
the binary system.(1 &
3)

By plotting the individual
orbits of the two stars in the system, the centre of mass of the
system can be determined.Then,
comparing the relative sizes of the two orbits around the centre of mass enables
the ratio of the massesm1 / m2 of the two stars
to be determined.Having both the
sum of the masses and the ratio of the masses then allows the calculation of the
individual masses of the two stars to be achieved.(1 & 3)See Ref. 1
pp.205-208 for a detailed analysis.

For spectroscopic binaries
the wavelength shift of each star’s spectral lines is measured and then
this data is used to calculate the radial velocity of each star from the
relevant Doppler shift formula.Remember
that radial velocity is velocity along our line of sight.This radial velocity data is then used to plot a radial velocity curve
(radial velocity versus time) for each star.The ratio of the masses of the stars in the binary is obtained
from this curve, along with the orbital period of the binary.

Kepler’s Third Law and Newtonian
mechanics are then used to relate the sum of the masses of the stars to
the orbital period of the binary, the orbital speeds of the stars and the angle
of tilt of the binary’s orbit to our line of sight.If all of these details are known, then again the individual
masses of the two stars can be determined.Often, however, the angle of tilt is not known accurately and therefore
stellar mass determinations from spectroscopic binaries are often uncertain.(1 & 3)See Ref. 1
pp.208-211 for a detailed analysis.

There is one case in which the
angle of tilt is known and that is when the spectroscopic binary is also
an eclipsing binary.The
orbital plane of the binary must then lie edge-on to our line of sight.Clearly, with this knowledge and with values for the orbital period of
the binary and the orbital speeds of the component stars, the sum of the
masses of the stars can be determined.Then, since both the sum of the masses and the ratio of the masses are
known, the individual masses of the two stars can be determined.(1 & 3)See Ref. 1
pp.211-214 for a detailed analysis.

Note that in the case of astrometric
binaries, it is usually possible to measure the orbital period of the binary
and the average distance of separation of the two stars.Thus, using Newton’s form of Kepler’s Third Law we can get an
estimate of the total mass of the binary system.It is usually not possible to determine the masses of the individual
stars.

Example
Questions

1.An astrometric binary is observed to have a period of 44.5 years and an
orbit with an average distance of separation between the component stars of 100
AU.Determine the sum of the masses
of the stars in this binary.(Data:
1AU = 1.5 x 1011 m)(Answer:
1.012 x 1033 kg)

2.An eclipsing binary has a period of 44.5 years and an average distance of
separation between the component stars of 3.9 AU.Determine the combined mass of the binary system.(Answer: 6.0 x 1028 kg)

CLASSIFICATION
OF VARIABLE STARS

A variable star is one that varies in brightness (5).As at January 2004, the total number of designated variable stars was
38622 according to the “General Catalogue of Variable Stars”, 4th
Edition, by P N Kholopov et al, Moscow (1985) and its published updates (also
available on the Internet).There
are two broad categories of variable star: (a) Extrinsic Variables, which
vary in brightness for some reason external to the star; and (b) Intrinsic
Variables, which vary in brightness due to changes in the star itself.Certain stars may vary in brightness due to both of these reasons. (5)

EXTRINSIC
VARIABLES:

As mentioned above, Extrinsic
Variablesvary in brightness for some reason external to the star.Such reasons include rotation, orbital motion or obscuration.Examples of common types of extrinsic variables include:

(a)Rotating variables, in which either the star’s ellipsoidal shape
or the presence of large cooler or hotter areas on the star’s surface change
the star’s brightness as it rotates.BY Draconis stars are examples of the latter.

(b)Eclipsing Binaries, which as discussed previously, consist of
orbiting stars that periodically eclipse each other.Algol (Beta Persei) is an example of an eclipsing binary system about 100
light years from Earth.

INTRINSIC
VARIABLES:

Intrinsic Variables vary in
brightness due to actual changes in the luminosity of the star itself not due to
external processes such as rotation or eclipses.The majority of variable stars fall into this broad category.Intrinsic variables are further sub-divided into non-periodic or
periodic variables.

NON-PERIODIC
VARIABLES:

These are intrinsic variables that
show irregular variations in brightness.Examples of such variables include:

(a)Novae – A nova is a star that undergoes a sudden, unpredictable
increase in brightness typically of 11 to 12 orders of magnitude (5).The sudden increase in brightness is accompanied by an explosive loss of
a relatively small amount (about 10-5) of the mass of the star.The outer atmosphere is blown away to form an expanding shell of gas
around the star that can often still be seen many years after the explosion.Nova Cygni, which flared up in 1975 by about 19 magnitudes is an
example. (10)

(b)Supernovae – A supernova is a violently exploding star, which
may become over a billion times brighter than the Sun, and for many weeks may
outshine the entire galaxy in which it lies (5).The explosion completely destroys the star and throws most of
its matter into space at high speeds, leaving behind either a tiny, very dense,
collapsed remnant called a neutron star or in some cases a black hole.SN1987A is a supernova observed in the Large Magellanic Cloud in
1987.The Crab Nebula is the
remnant of a supernova explosion in the constellation Taurus in 1054. (10)

(c)R Coronae Borealis Stars – These stars are sometimes referred to
as “reverse novae” since they decrease in brightness by as much as 10
magnitudes before returning to normal.They are supergiants with carbon-rich atmospheres.The sudden minima are due to the accumulation of clouds of carbon dust,
which are then blown away and allow the star to return to its normal brightness.The decreased brightness phase may last from months to years. (5 &
10)

(d)Flare Stars – These stars are red dwarfs that exhibit intense
outbursts of energy (flares) from small areas of their surface.Examples include UV Ceti stars and certain BY Draconis stars. (5)

(e)T-Tauri Stars – These are very young protostars.They are less than 10 million years old, with masses similar to or less
than that of the Sun and with diameters several times that of the Sun.They are still contracting.They
exhibit irregular variability ranging from ultraviolet flares on a time scale of
minutes to optical variations on time scales of days to years.The prototype, T Tauri itself, lies within Hind’s
Variable Nebula and varies irregularly between 8th and 13th
magnitudes. (5 & 3)

PERIODIC
VARIABLES:

These are intrinsic variables that
show regular variations in brightness.These variables are also known as pulsating variables.A pulsating variable undergoes a periodic cycle of contraction and
expansion as the star oscillates between a state where the inward acting
gravitational force is dominant and another where the outward acting radiation
pressure is dominant.The
periodic contraction and expansion produces the observed periodic variation in
brightness.As the star
contracts, the increase in luminosity caused by the temperature increase exceeds
the decrease caused by the contraction in radius, so the star becomes brighter.As the star expands, the luminosity and therefore the brightness
decrease. (9 & 10).

The types of periodic variables
can be categorized by the period and amplitude of their light curves.The period is simply the amount of time between one occurrence of
maximum (or minimum) brightness and the next.The amplitude is the difference in magnitude between maximum and
minimum brightness.Examples of
periodic variables include:

(a)Cepheid Variables – These are the best-known examples of
pulsating variables and are named after Delta Cephei, the first star of
this type to be discovered.Cepheids
are very luminous yellow giant or supergiant stars.Their brightness usually varies by about one magnitude with a
period of between 1 and 135 days, while their radius typically varies by 10 to
20 percent.We will say a little
more about these very important stars later.They are extremely useful as a means of distance measurement.(5 & 10)

(b)Mira Stars (also called Red Variables) – These are the most
common pulsating stars.They are
long period pulsating red giants and supergiants.They take their name from the star Mira (Omicron Ceti)
which lies about 250 ly from Earth.Mira
stars have periods of about 80 to 1000 days and have an amplitude of 2.0 to 10
magnitudes. (5 & 10)

(c)RR Lyrae – These are the second most common type of variable.They are old yellow giants.Most
have periods between 0.2 and 1.2 days and amplitudes of 0.2 to 2.0 magnitudes.One of the two types of RR Lyrae star (the RRAB type) all
have approximately the same absolute magnitude of +0.5, making them valuable
distance indicators. (5 & 10)

(d)RV Tauri – These are highly luminous, yellow supergiants.The light curves have overall amplitudes of 3 to 4 magnitudes.The period lies in the range from 30 to 150 days. (5)

NOTE:The above classification of variable stars is
the one required by the current HSC Astrophysics Syllabus.This classification scheme is based on the form of the light curve for
the star, its amplitude and periodicity or lack of it.This classification scheme has been superseded. I
do not know why the Syllabus Committee decided to go back to the old system.The new classification scheme adopted by the “General Catalogue of
Variable Stars”, 4th Edition, is based on the physical mechanisms
that underlie the different forms of variation or the physical structure of the
stars themselves.This new
classification scheme allocates variable stars to one of seven classes:
Eruptive, Pulsating, Rotating, Cataclysmic (explosive nova-like), Close Binary
Eclipsing Systems, Optically Variable X-Ray Sources and finally Unique Variables
(for mistakes and variables no-one has figured out yet).

SUMMARY
OF CLASSIFICATION OF VARIABLE STARS:

When trying to remember the
classification scheme for variable stars, a simple flow chart like the following
can be of great assistance.This
particular flowchart appears in Ref.12 on p.326.A similar one is shown in Ref.9 on p.46.

POSITION OF VARIABLE STARS
ON HR DIAGRAM:

The following figure shows where certain types of variable stars are
found on the HR Diagram.Cepheid
variables and RR Lyrae variables are located in the Cepheid
instability strip
(between the dotted lines on the diagram), which occupies a region between the
main sequence and the red giant branch.A
star passing through this region along its evolutionary track becomes unstable
and pulsates. (3)

The diagram above is an amalgamation of similar diagrams
from Ref. 3 p.530, Ref. 12 p.327 and Ref. 9 p.47.

PERIOD-LUMINOSITY
RELATIONSHIP

In 1912, American astronomer Henrietta
Leavitt reported her important discovery of the period-luminosity
relationship for Cepheid variables.Leavitt
studied numerous Cepheids in the Small Magellanic Cloud, a small galaxy near our
own Milky Way.Leavitt found that the
periods of these Cepheids were directly related to their average luminosities.The longer the Cepheid’s period, the greater is its luminosity. (3
& 9)

This period-luminosity
relationship is very important in astronomy because it can be used to determine
distances to objects in the universe (3).The
process is as explained below.

Cepheid variables vary in a
regular and characteristic way, increasing in brightness rather more rapidly
than they decrease (10).Cepheid
variables therefore have very distinctive light curves, which are plots
of brightness versus time.A
typical light curve appears as below.

From this light curve we can
determine both the period of the Cepheid variable under study and its average
apparent magnitude.In this case
the period is 8 days –the time between one occurrence of maximum
brightness and the next.The
average apparent magnitude is(4.0 + 3.5)/2 which equals 3.75.Using the period we can then determine the average absolute magnitude
of the Cepheid variable from the appropriate period-luminosity curve (see
the next graph).Once the average
absolute magnitude is known we can use the distance modulus formula to
calculate the distance to the required Cepheid variable.

Note that initially Leavitt and
others thought that all Cepheids could be described by the same
period-luminosity relationship.This,
however, was not the case.We now
know that there are two different types of Cepheid variable, each with its own
period-luminosity relationship.Type
I or Classical Cepheids are the brighter, more massive (5-15 solar masses),
metal-rich, younger, second generation stars, found exclusively in the disc
population of galaxies, where they are often members of open clusters (5, 10
& 12). Type II or W Virginis Cepheids are the dimmer, less massive
(0.4 to 0.6 solar masses), metal-poor, older, red, first generation stars (5, 10
& 12).Astronomers examine
a star’s metal content from its spectrum in order to classify it as either a
Type I or Type II Cepheid (3).The period-luminosity
relationship for both types of Cepheid variable is shown in the following
graph.

This graph is an adaptation of the
graph shown on p.49 of Ref.9.Note
that the relationship for Type I Cepheids has only been drawn up to a period of
50 days.In reality, Type I
Cepheids can have periods up to 135 days.Remember
too that the straight lines in the graph are simply showing general trends that
are seen in the data.The actual
period-luminosity graphs recorded by astronomers consist of bands of stars
plotted on and around these lines.See
Ref.1 Fig.14.4 on p.545 for example.

Now continuing with our example
of how to determine the distance to a Cepheid, let us assume that the Cepheid in
question has been identified as a Type I Cepheid.So, using the known period of 8 days, we can read off the
period-luminosity relationship that the absolute magnitude of our Cepheid is
–2.5.

The final step is to use the
distance modulus formula to calculate the distance to the Cepheid.

So,
on re-arrangement we have:

which yields d = 177.8
parsecs, when m
= 3.75 and M =
-2.5 are substituted into the above equation.Thus, the distance to our Cepheid variable is 177.8pc.

As mentioned previously, this
method of using the period-luminosity relationship for determining distances to
Cepheids is very important in astronomy.It
provides a relatively simple and accurate method of distance calculation.An astronomer identifies a Cepheid as either Type I or Type II from its
spectrum.He/she then records the
light curve for the Cepheid and from this determines the period and average
apparent magnitude of the Cepheid.Using
the period, the astronomer then determines the absolute magnitude of the Cepheid
from the period-luminosity relationship and finally calculates the distance to
the Cepheid using the distance modulus formula. (3)

Cepheid variables are extremely
luminous objects.They can be seen
even at distances of millions of parsecs.By
applying the period-luminosity relationship in this way to Cepheids in other
galaxies, astronomers have been able to calculate the distances to those
galaxies with great accuracy.Such
measurements play an important part in determining the overall size and
structure of the universe. (3)

THE
PROCESSES INVOLVED IN STELLAR FORMATION

Where in our galaxy does star
formation occur?The spiral
arms of our Milky Way galaxy are laced with giant molecular clouds (GMC),
immense clouds of interstellar gas (mainly hydrogen) and dust, so cold that
their constituent atoms can form into molecules (3).Particularly dense regions within these clouds form what are called “Dark
Nebulae” and within these, stars are formed.Star formation commences in these dark nebulae when a GMC is
compressed, which can happen as the cloud passes through one of the spiral arms
of the galaxy or as a result of a nearby supernova explosion (3).

A nebula is an interstellar
cloud of gas and dust.A dark
nebula is a nebula with a relatively dense concentration of microscopic dust
grains, which scatter and absorb light very efficiently.The Horsehead Nebula in the constellation of Orion is a very
good example of a dark nebula.Dark
nebulae partially or completely block our view of any stars that lie behind
them.A typical dark nebula has a
low temperature of 10 K to 100 K, which is low enough for hydrogen to form
molecules, and contains from 104 to 109 particles (atoms,
molecules & dust grains) per cubic centimetre.(3)

These characteristics make dark
nebulae the only parts of the interstellar medium suitable for star
formation.The relatively high
density enhances gravitational attraction within the medium and the low
temperature ensures a low (outwards pushing) pressure within the medium.(3)

Within a dark nebula, the densest
regions of gas and dust may begin to contract under their own gravity.This contraction may also be triggered by the
gravity from a passing star or the shock wave from a nearby supernova explosion.The density at the centre of a contracting region increases more quickly
than that at the outer edges.Thus,
the gravitational attraction at the centre increases more quickly than that at
the edges, resulting in the formation of a central core of material surrounded
by a more slowly contracting envelope.(3
& 9)

As the core contracts, the gravitational potential energy of its
constituent particles is transformed into kinetic energy, heating the core.This heat causes the gas in the core to glow.Convection currents carry the heat outwards, creating an outward-directed
pressure that opposes the gravitational contraction of the core.As the core temperature increases, the outward pressure increases until
eventually the gravitational collapse of the core is almost balanced by the
outward pressure.At this stage the
core is called a protostar.Protostars
with masses between 0.08 solar masses and 100 solar masses will eventually
evolve into main sequence stars. (1 & 3)

It is worth noting that even though young protostars are quite luminous,
they cannot be detected using visible light.This is because the dust in the protostar’s immediate surroundings,
called its cocoon nebula, absorbs much of the visible light emitted by
the protostar.However, as the dust
in the cocoon nebula re-radiates this absorbed heat at infrared wavelengths, the
presence of the protostar can be detected using infrared telescopes.Many protostars have been detected in this way.(3)

During the birth process stars both gain and lose mass.Mass loss occurs in a couple of different ways.Protostars less than 3 solar masses go through a T-Tauri phase in which
they eject up to a solar mass of material over about 107 years and
exhibit irregular variation in luminosity on a time scale of a few days.Protostars more massive than 3 solar masses do not vary in luminosity
like T-Tauri stars but do lose mass due to the huge radiation pressure at their
surfaces that blows gas into space.Many
protostars also lose mass by ejecting gas along two oppositely directed jets in
a process called bipolar outflow.The
huge stellar winds produced by these mass ejection processes often blow away the
remnants of the surrounding cloud and allow the protostar to be seen in visible
light.(3)

Protostars gain mass at the same time as they are losing it.As the envelope of material around the protostar contracts, it spins
faster and faster and flattens into a disk with the protostar at the centre.Particles orbiting the protostar within the disk collide with each other,
causing them to lose energy and spiral inward onto the protostar, adding to its
mass.This process is called accretion.(3)

While these processes of mass loss and gain are continuing, so too is
the very slow contraction of the protostar itself.There is still insufficient energy flowing outwards from the
protostar to completely balance the inward pull of gravity.As the radius of the protostar decreases, its luminosity decreases and
its internal temperature increases.Eventually,
the internal temperature of the protostar reaches around 107 K,
sufficient for thermonuclear reactions to begin converting hydrogen to helium.These reactions eventually produce sufficient heat and
internal pressure to stop the star’s contraction.The outward radiation pressure and gas pressure has balanced the
inward force of gravity.Hydrostatic
equilibriumhas been achieved.Also,
the rate at which energy is produced in the core has balanced the
rate at which energy is transported to the surface of the star and radiated away
into space.Thermal equilibrium has
been established.The
protostar is now said to be a zero-age main sequence star and the processes of
stellar formation are complete. (3)

Note that the dark nebulae in which star formation occurs typically
contain tens or hundreds of solar masses of gas and dust, enough to form many
stars.Thus, young stars tend to
form in groups or clusters.Star
clusters typically include stars with a range of different masses, all of which
began to form out of the parent nebula at roughly the same time.(3)

PRE-MAIN-SEQUENCE
EVOLUTIONARY TRACKS

Curves that depict the life histories of stars on the H-R Diagram are
called evolutionary tracks (1).The
diagram below shows the evolutionary tracks of five protostars of different
masses.Each track shows us how the
protostar’s appearance changes in terms of its luminosity and temperature
because of changes in its interior.Where on the main sequence a given track ends depends on the
mass of the protostar.Protostars
eject a lot of mass into space as they form, so the mass shown for each
evolutionary track is the mass of the final main sequence star.Note that for a star like the Sun, the birth process takes
about 50 million years.(This
diagram was produced by Prof. Dale Gary of the New Jersey Institute of
Technology. The original link was: http://physics.njit.edu/~dgary/202/Lecture18.html.
This link no longer works.)

Note that the plot of the main
sequence using only stars of zero-age is called the zero-age main
sequence (ZAMS) plot (12).It
forms the complete diagonal main sequence shape shown on most generic H-R
Diagrams (12).Stars on ZAMS
have just ended their protostar stage (3).

The theory of how protostars
evolve helps explain why the main sequence has both an upper and lower mass
limit.A protostar of less than
0.08 solar masses can never develop sufficient pressure and temperature to start
hydrogen fusion in its core.Such a
protostar becomes a hydrogen-rich brown dwarf – a failed star.A protostar more massive than 100 solar masses rapidly becomes very
luminous, resulting in tremendous internal pressures.These pressures overwhelm gravity, expel the outer layers into space and
disrupt the protostar.(3)

LIFE
ON THE MAIN SEQUENCE

By definition, a main
sequence star is one that produces energy by the fusion of hydrogen nuclei
(protons) to helium nuclei in its core.This fusion reaction produces energy by the conversion of some of the
hydrogen nuclei mass into energy according to Einstein’s equation, E = mc2.Note that astronomers are notorious for referring to fusion reactions as “burning”.So they speak of “hydrogen burning” instead of hydrogen
fusion and “helium burning” instead of helium fusion, and so on.

Two different fusion
mechanisms are responsible for the helium production and consequent release
of energy in main sequence stars.Both
mechanisms can occur simultaneously in a main sequence star.However, for stars whose core temperatures are below 16
million K the proton-proton chain reaction is the main mechanism, while for
stars whose core temperatures are above this, the carbon-nitrogen-oxygen (or CNO)
cycle predominates (3).Let us
now have a brief look at these two mechanisms.

This reaction predominates in
stars like our Sun.Originally
proposed by the American physicist Charles Critchfield, this reaction has three
branches (3).Since the primary
branch PP I, accounts for the production of 85% of the Sun’s energy, we will
consider only this branch of the reaction in detail.PP I consists of three steps:

For those not familiar with
nuclear equations, refer to the Key above.In step 1, two protons combine to form a deuterium nucleus (an isotope of
hydrogen), a positron and a neutrino.In
step 2, another proton combines with the deuterium nucleus to form a nucleus of
light helium and a gamma ray photon, which carries energy away from the
reaction.In step 3, two light
helium nuclei combine to produce a nucleus of ordinary helium and two protons.(10)

Thus, the overall reaction is to
convert four protons into a nucleus of helium with the release of some energy.

In the PP II and PP III branches
of the reaction the light helium produced in step 2 above suffers different
fates.Details of these branches
can be found in Refs. 1 & 3.

This reaction mechanism
predominates in stars whose core temperatures are above 16 million K.Hans Bethe and Carl von Weizsacker discovered it
independently (3).In the CNO cycle
the carbon-12 nucleus acts as a catalyst and the following six-step
reaction takes place (10).

Overall, in this reaction four
protons are converted into a helium nucleus, two positrons, two neutrinos and
high-energy gamma ray photons (10).Again,
for those not familiar with nuclear reactions the following Key is provided.

For the CNO cycle to proceed,
there must be carbon-12 nuclei present.Obviously,
as the carbon-12 is returned at the end of the cycle, it is not actually used in
by the reaction.

Throughout the lifetime of the
main sequence star, the helium produced by the proton-proton chain reaction
and CNO cycle accumulates in the centre of the star,since it is denser
than the hydrogen.Also as the main
sequence star ages, changes occur in its luminosity, surface temperature and
radius (3).Hydrogen burning
decreases the total number of atomic nuclei in the star’s core (four hydrogen
nuclei are used up to make each single helium nucleus).The resulting decrease in internal pressure causes the core to contract
slightly under the weight of the star’s outer layers.In turn, this contraction increases the core’s density and temperature,
which effectively raises the pressure in the core to a level higher than it was
previously.

The increased core pressure
pushes outwards on the star’s outer layers, causing the star’s radius to
increase slightly.Also, the
increased density and temperature in the core cause hydrogen nuclei in the core
to collide more frequently, causing the rate of hydrogen burning to increase.Hence the star’s luminosity increases.Since the star’s surface temperature depends on the star’s luminosity
and radius, it changes as well.Thus,
as the star ages, its core shrinks and its outer layers expand and shine more
brightly.As an example, over
the last 4.6 x 109 years, our Sun has become 40% more luminous, grown
in radius by 6% and increased its surface temperature by 300K.(3)

As a main sequence star ages and
evolves, the increase in energy outflow from its core also heats the material
immediately surrounding the core.As
a result, hydrogen burning can begin in this surrounding material.This is called shell hydrogen burning since it is happening in the
shell surrounding the core.By
tapping this fresh supply of hydrogen, a star manages to last a few extra
million years on the main sequence.(3)

A star’s lifetime on the main
sequence depends critically on its mass.The more massive the star, the shorter its main sequence lifetime.This is because the more massive the main sequence, the more luminous it
is.In order to emit energy so
rapidly, these massive stars deplete their hydrogen stocks very much more
quickly than less massive stars.High
mass O and B stars completely exhaust their hydrogen supplies in only a few
million years, whereas low mass M stars take billions of years to use all their
hydrogen.A star of around one
solar mass will spend roughly 1010 years on the main sequence.So, our Sun, which has been on the main sequence for about
4.6 billion years, should have about another 5 billion years left to enjoy the
main sequence status.(3)